Symbolic Inverse Laplace Transform Applet
Rules and Theories
A rational polynomial is defined by where the numerator N(s) and the denominator M(s) are polynomials with real-valued coefficients. For practical purposes, the degree of N(s), n, is assumed to be less than that of M(s), m. The key to perform the partial fraction expansion is the find the m poles. Denote to be the j-th distinct complex root of M(s) with multiplicity . Then
According to Heaviside's expansion theorem,
The challenges are:
The above issues are handled with Java's OOP approach.
The Applet and User's Guide
In the Laplace transform domain, suppose X(s), Q(s), and Y(s) represent the input signal, the system transfer function and the output signal, respectively. In general, the input/output relation is H(s) = Y(s) = X(s) Q(s). This applet can be used to find the time-domain response y(t) for any known input signal and transfer functions, provided Y(s) can be expressed in form of a rational polynomial. There are two notable special cases: (i) For an impulse excitation, X(s) = 1, then letting H(s) = Y(s) = Q(s) will lead to the result of the impulse response. (ii) For a step function input, X(s) = 1/s, then letting H(s) = Q(s)/s will lead to the step-function response.
Validation with Known Transform Pairs
This applet has been tested with selected examples for perfect agreement..
M. E. Van Valkenburg, "Network Analysis", 2nd Edition, Prentice Hall, Englewood Clifs, 1964.
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